Seminar in Geometric Function and Mapping Theory

Thursdays 13:00–14:30

Institute of Mathematics of the Polish Academy of Sciences
ul. Śniadeckich 8, Warsaw, room 6 (regular meetings)

Organizers: Tomasz Adamowicz, Damian Dąbrowski

Upcoming seminars

22 October

Fernando Ballesta Yagüe (Universidad Complutense de Madrid)

Motivating weighted theory problems through boundary value problems

Abstract: In this talk, we will explain how boundary value problems for the Laplacian in planar Lipschitz graph domains can be studied through weighted theory. We will consider boundary data satisfying certain integrability conditions, like Lp or Lp,1 , and we will impose different boundary conditions, like Dirichlet, Neumann or a directional derivative.

The technique, introduced by Carlos Kenig in the 80’s, uses a conformal mapping to transfer the problem to the upper half-plane. There, the problem can be treated using Harmonic Analysis techniques. We will emphasise what kind of weighted theory problems are motivated by this approach.

This is joint work with María J. Carro and Virginia Naibo.

5 November

Julian Weigt (ICTP, Trieste)

TBA

Abstract: TBA

19 November

Iván Caamaño Aldemunde (Universidad de Granada)

TBA

Abstract: TBA

26 November (14:00)

Marcin Gryszówka (Uniwersytet Warszawski)

TBA

Abstract: TBA

3 December

Giacomo del Nin (Max Planck Institute for Mathematics in the Sciences, Leipzig)

TBA

Abstract: TBA

10 December

Wojciech Górny (University of Warsaw)

TBA

Abstract: TBA

Past seminars (2026/2027)

15 October

Tommaso Bertin (IMPAN)

Lavrentiev Phenomenon and integral representation of the lower semi-continuous envelope for functionals with non convex and non continuous Lagrangians

Abstract: In this seminar, we investigate classical integral functionals in the Calculus of Variations and study conditions on the Lagrangian that guarantee the absence of the Lavrentiev Phenomenon. In particular, we focus on Lagrangians that are non-convex, possibly highly discontinuous, and unbounded with respect to the gradient variable.

Our first step to prove the non-occurrence of the Lavrentiev Phenomenon is to prove the representation of the lower semicontinuous envelope of a functional defined in W1,∞(Ω) as an integral functional whose Lagrangian is given by the bipolar of the original one.

Specifically, we adapt a technique presented in the classical book by Ekeland and Temam for the bounded case, and a refinement of a method due to Cellina, for the unbounded case. The integral representation also allows us to apply recent results by Bousquet, Mariconda and Treu to the non-convex setting.

We establish the integral representation of the lower semicontinuous envelope and the corresponding absence of the Lavrentiev Phenomenon under very weak assumptions in the autonomous case and under suitable anti-jump conditions on the spatial variable in the non-autonomous case. Furthermore we prove the strong convergence of the approximating sequence for minimizers or under specific growth assumptions on the Lagrangian.

These results provide new insights into the interplay between regularity assumptions, relaxation methods, and the avoidance of the Lavrentiev Phenomenon, thereby extending recent advances to a broader class of variational problems.

This is a joint project with with G. Treu (Università di Padova) and P. Huguet (université de Toulouse III Paul Sabatier).

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